Yuan’s componentwise rigidity conjecture

Let $n\geq 2$, let $\mathbb{B}^n$ be the complex unit ball with its canonical Kähler–Einstein metric $g_{\mathbb{B}}$, and let $\Omega_1,\ldots,\Omega_m$ be irreducible bounded symmetric domains with canonical Kähler–Einstein metrics $g_i$. If $F=(F_1,\ldots,F_m):\mathbb{B}^n\to\Omega_1\times\cdots\times\Omega_m$ is a holomorphic isometry for the product metric $\sum_{i=1}^m c_i g_i$, where $c_i>0$, then every nonconstant component $F_i$ is a holomorphic isometry up to a positive constant factor: there exists $\lambda_i>0$ such that $F_i^*g_i=\lambda_i g_{\mathbb{B}}$.